Theoretical study of the binding of Na clusters encapsulated in the ...

PHYSICAL REVIEW B

VOLUME 53, NUMBER 23

15 JUNE 1996-I

Theoretical study of the binding of Na clusters encapsulated in the C240 fullerene J. M. Cabrera-Trujillo Facultad de Ciencias, Universidad Auto´noma de San Luis Potosı´, 78000 San Luis Potosı´, Mexico

˜iguez, M. J. Lo´pez, and A. Rubio J. A. Alonso, M. P. In Departamento de Fı´sica Teo´rica, Universidad de Valladolid, 47011 Valladolid, Spain ~Received 17 January 1996! Density functional theory has been used to study the electronic structure and binding of Na N clusters (N240) ~Ref. 31! and potassium and hydrogen endohedrals in C 60 . 22 We first give a brief sketch of the model and computational method, which are presented in Sec. II. The results are presented in Sec. III, which is divided in two parts. In the first part, we describe the results for rigid Na clusters inside the fullerene C 240 . This allows us to describe the trends in the electronic structure while, in the second part, we present the results obtained by allowing for the relaxation of the Na atoms. In both cases, the C 240 fullerene is assumed undeformable, that is, the carbon atom positions are always held fixed. Finally, the concluding remarks are presented in Sec. IV. II. MODEL

For a given cluster geometry characterized by the set of ionic positions $ Rj‰, the ground-state energy is obtained by minimization of the energy density funcional:27,28 E @ r # 5T s @ r # 1

1 2

EEr

1E xc@ r # 1

~ r! r ~ r8! drdr81 u r2r8u

Er

~ r! V ions~ r! dr

1 U ~ u Rj2Rj8u ! , 2 jÞ j 8

(

~1!

where the first term stands for the independent-particle kinetic energy, the second is the classical electrostatic energy of the electrons, the third term is the electron-ion interaction, the fourth term includes the exchange and correlation energies, and the last one takes account of the ion-ion repulsion energy. Atomic units are used through this paper unless otherwise stated. The variation of E @ r # , with respect to the electronic density r (r), leads directly to the Kohn-Sham28 equations

F

2

G

¹2 1V eff~ r! c i ~ r! 5« i c i ~ r! , 2

~2!

for the single-electron orbitals c i (r). These equations are solved self-consistently for the valence electrons moving in the effective potential V eff . The electronic density r (r) is obtained from the occupied single-particle orbitals:

of Gunnarsson and Lundqvist was used for V LDA xc (r) ~Ref. 32!. The potential of each ion is modeled by an empty-core pseudopotential,33 where the attraction of the valence electrons by the ion is purely Coulombic outside a sphere of radius r c ~the empty-core radius! and zero inside. Thus, the total potential at some point r of the cluster, due to ions at positions $ Rj‰, is V ions~ r! 5

(j V ~ u r2Rj u ! .

~5!

This is a complicated three-dimensional potential. The problem can be reduced to a system of interacting electrons moving in an external potential well with spherical symmetry by using the SAPS model,29 that is replacing V ions(r… by its spherical average around the cluster center ~in our case, the center of symmetry of C 240): V ions~ r! →V SAPS ions ~ u ru ! .

~6!

Then, the process of integrating the Eqs. ~2!–~4! is a much easier one. The last term in Eq. ~1! represents the repulsion between pointlike ionic charges, placed at the positions $ Rj‰. This term is, of course, sensitive to the actual threedimensional geometry of the cluster. An additional approximation has been considered for the carbon atoms. As we explained above, the Eqs. ~2!–~4! are self-consistently solved under the assumption of spherical symmetry for the external potential. On the other hand, it is well known that the high stability of the fullerenes is due to three of the four valence electrons of the carbon atoms, which form strong directional s bonds with the three neighbor atoms, while the last electron forms weaker p bonds, which are delocalized over the fullerene cage. The delocalized character of p electrons plays a special role in the collective optical excitations22 and also in the chemical behavior of fullerenes interacting with impurities, similar to the interactions taking place in intercalated atoms in graphite. In our case, the interaction with the endohedral cluster again involves only the p electrons. Then we consider the carbon atoms as monovalent and with an ionic pseudopotential characterized by an empty core radius r c 50.21 a.u., obtained by fitting the first ionization potential of the free atom to experiment. In a similar way, the fitted core radius of the Na pseudopotential takes a value 1.74 a.u., which is greater than the core radius of the C atom, so that Na atoms are less attractive for the electrons than C atoms ~a higher core radius gives a larger volume where the attractive potential is canceled!, consistent with the higher ionization potential of C. III. RESULTS

occ

r ~ r! 5

53

( u c i ~ r! u . 2

i51

~3!

The self-consistent effective potential is given by V eff~ r! 5V H ~ r! 1V LDA xc ~ r ! 1V ions~ r ! ,

~4!

where the three terms on the right-hand side stand for the Hartree contribution, the exchange-correlation potential in the local density aproximation ~LDA!, and the external potential due to the ion cores, respectively. The approximation

We divide our study of endohedral Na N clusters in two steps. In the first step, the atomic structure of the NaN cluster inside C 240 is assumed the same as in the free Na N cluster. For this reason it is useful to consider previously the electronic and atomic structures of isolated C 240 and Na N clusters. In a second step, the Na N cluster will be relaxed to reach its equilibrium configuration inside the rigid fullerene. Comparison of the electronic structure and energies in the two situations ~relaxed and unrelaxed endohedral! allows us

53

THEORETICAL STUDY OF THE BINDING OF Na CLUSTERS . . .

to understand the changes induced by the interaction with the enclosing cage. The ground-state geometries of free Na N clusters have been previously obtained within the same formalism.29 One of the main results of those calculations is the development of atomic shells ~layers!: for small N, most of the atoms form a single atomic shell and, as the cluster grows, atoms begin to occupy the inner region inside this shell. More precisely, for N between 9 and 19 there is one single atom inside the cluster, all the rest forming the surface. From N5 20 and up to 30 atoms, the cluster can be viewed as formed by two atomic shells, and the inner one has a population ranging from 2 to 5 atoms. It is also well known that Na N clusters with N52, 8, 18, and 20, are specially stable, due to the closing of electronic shells. The DFT-SAPS calculation29 gives results in agreement with the observed enhanced abundance of such clusters in the experimental mass spectra.34 Turning to C 240 , we model this fullerene by a spherical cage where the carbon atoms are distributed in a similar way as in the corresponding truncated-icosahedron fullerene; each carbon atom is placed on the vertices of slightly distorted pentagonal and hexagonal rings distributed on a spherical hollow cage with radius R513.46 a.u. This value has been taken from ab initio DFT calculations,35 which show that the spherical shape is the most stable geometry adopted by giant fullerenes. The self-consistent electronic charge is confined to a rather thin spherical shell.31 The electronic structure of C 240 in the SAPS model is 1s 2 1p 6 1d 101 f 141g 181h 221i 261 j 301k 341l 381m 40 ~the occupation of the shells are given as superscripts!,31 and corresponds to the filling of shells up to a maximum angular momentum equal to 10. The orbitals have principal quantum number n51, that is, those are orbitals without nodes. The uppermost electronic shell of C 240 can be completed with two additional electrons, in which case the fullerene becomes a more stable ionized species. The SAPS calculation of the electron affinity, defined as A~C 240)5E(C 240)2E(C 2402 ), gives the value A(C 240)53.81 eV. On the other hand, Na N clusters can donate electrons easily, because of their small ionization potentials. For this reason, we can expect that one or more electrons will be transferred from the Na N cluster to the fullerene cage.

16 061

FIG. 1. Structure of the rigid Na 13 cluster encaged by the spherical fullerene C 240 . The big and small circles denote Na and C atoms, respectively.

but this is not the case for the energy of the system, which contains an ionic-repulsion contribution, which depends on the precise positions of the ions. Optimal orientations will result from the relaxation process, along with relaxed radial distributions of the Na atoms. We postpone the discussion of the binding energy until Sec. III B and restrict the discussion here to the electronic structure for the unrelaxed case. The electronic structure of C 240 was described above. In the self-consistent calculation for Na N @C 240 , electronic shells appear with orbitals having one node (2s, 2p, 2d, and 2 f ) or two nodes (3s). The effective potential V eff and the calculated one-electron energies of the occupied shells are plotted in Figs. 2 and 3 for Na N @C 240 with N54 and 20, respectively. The orbital energies are represented by horizontal lines. The different shells can be identified from the electronic configurations given on the bottom part of the figures.

A. Unrelaxed Na N @C 240

We begin the study of the interaction between the endohedral clusters and the cage by using an approximation in which both the Na and C atoms are frozen at the same positions as they have in the pure Na N clusters and the C 240 fullerene, respectively. The endohedral Na N is placed such that its center of mass coincides with the center of C 240 and the pseudopotential V ions(r… is spherically averaged around this point. Figure 1 shows Na 13@C 240 as an example. We do not consider off-center endohedrals, because our spherically averaging procedure is not adequate for such a situation, although for the smaller sizes it is plausible that the binding to the cage may increase in an off-center position. The electronic structure is independent on the relative orientation of the endohedral Na N cluster and the fullerene, due to the spherical averaging of the ionic potential in the SAPS model,

FIG. 2. Total electron density r (r) and partial contributions from the uppermost levels of Na 4 @C 240 . The effective potential V eff(r) and the occupied energy levels ~drawn as horizontal lines! are shown on the lower part. The electronic configuration of the system is given at the bottom.

16 062

J. M. CABRERA-TRUJILLO et al.

FIG. 3. Same as Fig. 2, but for Na 20@C 240 .

The total electron density is also plotted as well as some partial electron densities corresponding to the occupied levels near the highest occupied molecular orbital ~HOMO!; these partial densities are relevant for discussing the bonding between the endohedral and the cage. The radius of C 240 is R513.46 a.u. whereas the most external atomic layers of those two Na clusters have radii equal to 3.95 and 8.0 a.u., respectively. The effective potential of Na 4 @C 240 is formed by two potential wells corresponding to Na 4 and C 240 , respectively, separated by a potential barrier. Focusing on the outermost occupied shells, the 2s orbitals are localized on the inner region, whereas the 1k, 1l, and 1m orbitals are localized on the region corresponding to C 240 . The 1m orbital, which is the HOMO of the whole system, is filled with 42 electrons. This indicates that two electrons are transferred from the p shell of the Na 4 cluster to the cage, and this results in closed-shell configurations for both the endohedral, Na 4 21 , and the cage C 24022 . The ionic picture is a good description, because the electrons are well localized on either side of the potential barrier. In Na 20@C 240 there is again a nominal transfer of two electrons from the endohedral to the cage. However, the 2 p, 1m, and 2d shells have orbital eigenvalues above the maximum of the barrier ~and that of the 1l shell is nearly equal to the barrier maximum!. These orbitals overlap, giving rise to a covalent contribution. The conclusion from Figs. 2 and 3 is that, as the size of the Na N endohedral increases, the binding changes gradually from purely ionic ~at small N) towards a mixture of ionic and covalent bonding. A full view of the evolution of the energy eigenvalues corresponding to the electronic shells near the HOMO is given in Fig. 4. Up to N522, two electrons are transferred to the cage, except for the N values marked by the circles, where only one electron is transferred. At N523, the 1n level of C 240 begins to be filled with electrons coming from the 2 f level of the pure Na N cluster, whereas this 2 f level begins to be filled at N525 and the two levels 1n and 2 f become degenerate for N.25. The 2 f orbital overlaps substantially with the cage orbitals. The largest size oscillation in the orbital eigenvalues of Fig. 4 corresponds to the drop of the 2s level between N53, in which the Na 3 endohedral is single ionized, and N54, in which Na 4 is doubly ionized, so this large drop is caused by the extra attractive force exerted

53

FIG. 4. One-electron Kohn-Sham eigenvalues of the rigid endohedral Na N @C 240 as a function of N. The open circles mean that only one electron has been transferred to the 1m shell. All the other clusters have a complete 1m shell with 42 electrons, and for the larger sizes also the 1n shell has one or two electrons.

by the Na 4 21 molecular ion on the 2s electrons. Similar drops of the 2s and 2 p eigenvalues between N59 and N510 are hidded to some extent by the drop at N59, coming from the geometrical stabilization of Na 9 , due to the presence of the ninth Na atom at the cluster center. On the other hand, the perturbation by the endohedral cluster of the energy levels associated with the cage consists of a small uniform downward shift as N increases. The net shift between N51 and N530 is about 0.3 eV and the slope of the eigenvalue curves changes at N524. A way to analyze the bonding is by considering the density difference: D r ~ r ! 5 r s~ r ! 2 r ~ r ! ,

~7!

where r (r) stands for the calculated electron density of the composite system Na N @C 240 , and r s (r) is the reference density obtained by superposition of the electron densities of the separated components, Na N and C 240 . Figure 5 gives the

FIG. 5. Charge density difference D r (r) between the composite system Na N @C 240 and the superposition of the separated components. The curves are identified by the number N of Na atoms. Positive values of D r (r) indicate charge depletion.

53

THEORETICAL STUDY OF THE BINDING OF Na CLUSTERS . . .

behaviour of 4 p r 2 D r (r) for several values of N. This function shows oscillations that extend up to the external surface. A positive value indicates charge depletion in that region. The structure of all the curves is nearly the same: a region of charge depletion in the inner part, corresponding to the loss of charge by the Na endohedral, followed by a region of charge accumulation. The charge accumulated on the carbon cage is actually displaced a little towards the region between the endohedral and the cage. This is revealing the partially covalent nature of the bonding. Furthermore, there is also a region of charge depletion on the outerpart of the cage; this shows that also the cage orbitals are pushed a little towards the endohedral to increase the covalent contribution to the bonding. By integrating D r (r) from r50 up to the edge of the endohedral region where D r (r)50 ~this edge occurs at 10–11 a.u.!, we can get an estimate for the charge lost by the Na cluster. For instance, we find that the charges lost by Na 4 and Na 10 are 1.96e and 1.71e, respectively, which is consistent with our previous analysis of the electron configurations. For other values of N,10, an amount of charge very close to one electron is transferred, which is reflected in the smaller amplitudes of the density difference curves. The amplitude of the oscillations in the endohedral and bonding regions is nearly constant for N510 – 22, but for N>23, the amplitude increases progressively, reflecting the fact that more than two electrons are transferred to the cage. When N increases in the whole range of values studied, that is, from N51 to N530, the charge depletion on the outer surface of the fullerene becomes more and more important and also contributes to an increasing amplitude of D r (r) in the middle region, in particular, again, for N.20. We induce from this analysis a predominantly ionic bond reflected in the shell occupations and also in the charge density difference, with a partially covalent character coming from the charge accumulated in the overlapping region. The covalent contribution increases with the cluster size, and its associated charge screens the ionic attraction between the two ionized fragments. B. Relaxed clusters

Next, we perform a steepest-descent relaxation of the Na N cluster inside the cage starting from the geometries corresponding to the unrelaxed case. In this process, the carbon atoms are held at the same positions they have in the isolated C 240 fullerene, and the center of mass of the relaxed Na N group is maintained at the fullerene center. We expect that the assumption of a rigid cage is a reasonable approximation, because of the strong s bonds between the carbon atoms ~this point is treated in more detail below!. For the dimer, the Na-Na bond is stretched by 1 a.u. In this case, one electron is transferred to the 1m shell of the cage, just as in the unrelaxed case, and the repulsion between the two Na 1 ions, only partially screened by a single electron, is responsible for the increase in the Na-Na bond length. We obtain, in general, cluster structures formed by atomic shells similar to those of the free Na N clusters, although with expanded radii. Figure 6 shows the mean radius of the outer shell as a function of N 1/3. The outer shell radius of the relaxed cluster is larger than in the free clusters and the maximum expansion occurs around N513, for which the

16 063

FIG. 6. Mean radius of the external atomic shell of Na N clusters versus N 1/3 in the relaxed ~circles! and unrelaxed ~squares! cases.

relaxed and unrelaxed radii differ by 2.2 a.u. This expansion arises from two effects. One is the ionicity of the endohedral, already discussed for Na 2 . The second effect is that by expanding the radius of the endohedral, the outer orbitals overlap more effectively with those of the cage, optimizing the covalent contribution to the bonding. Between N513 and N530 the radius of the endohedral cluster increases slightly and soon reaches an almost constant value, in contrast with the linear increase exhibited by the free Na N clusters. For the largest sizes studied here, N>26, the expansion with respect to the unrelaxed case is negligible. Figure 7 shows the evolution of the Kohn-Sham singleparticle energy eigenvalues. There are some changes with respect to the unrelaxed case ~see Fig. 4!. Two regions can be distinguished in this plot. The first is that of N